Classification of knotted tori in the 2-metastable dimension

Details Classification of knotted tori in the 2-metastable dimension Classification of knotted tori in the 2-metastable dimension

2008-11-17

arxiv.org/abs/0811.2745v2

Mathematics

Geometric Topology

in English and in Russian, 23 pages, 10 figures

Scientific paper

This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings N->S^m. We study the specific case of knotted tori, i. e. the embeddings S^p x S^q -> S^m. The classification of knotted tori up to isotopy in the metastable dimension range m>p+3q/2+3/2, p2p+q+2. Then the set of smooth embeddings S^p x S^q -> S^m up to isotopy is infinite if and only if either q+1 or p+q+1 is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new beta-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement.

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